Göttingen-Hannover Number Theory Workshop 2014
- Date: 21 November 2014 (Friday)
- Location: Room A 410, Institut für Algebra, Zahlentheorie und Diskrete Mathematik, Leibniz Universität Hannover, Welfengarten 1, 30167 Hannover (4th floor of the main building Welfenschloss - travel information, Google Maps)
- Organizers: Valentin Blomer (Göttingen), Jörg Brüdern (Göttingen), Ulrich Derenthal (Hannover)
|Friday, 21 November 2014 (room A 410)|
|12:30-14:00||Lunch (Shalimar, Lange Laube 13 - from Hauptbahnhof: ~10 minute walk)|
|14:00-15:00||Péter Maga (Göttingen): Subconvexity for supnorms of automorphic forms on PGL(n)|
|15:45-16:45||Christopher Frei (Hannover): Forms of differing degrees over number fields|
|17:00-18:00||Oscar Marmon (Göttingen): Random Thue and Fermat equations|
- Péter Maga: Subconvexity for supnorms of automorphic forms on PGL(n)
As it was proved by Sarnak, the supnorm of eigenfunctions of the Laplacian on a compact symmetric Riemannian manifold can be estimated from above by an appropriate power (given in terms of some invariants of the space) of their Laplace eigenvalue. Examples show that Sarnak's exponent is sharp in some cases. However, when the space has also arithmetic symmetries (i.e. Hecke operators) and we restrict to joint eigenfunctions of the Laplacian and the Hecke operators, one might expect a better exponent. We prove that a better exponent exists for automorphic forms on PGL(n,R). Joint result with Valentin Blomer.
- Christopher Frei: Forms of differing degrees over number fields
Consider a system of m forms of degree d in n variables over the integers. A classical result by Birch uses the circle method to provide an asymptotic formula for the number of integer solutions to this system in a homogeneously expanding box, as long as n is large compared to m and d. An analogous result over arbitrary number fields was proved by Skinner. In joint work with M. Madritsch, we extend Skinner's techniques to a recent generalization of Birch's theorem by Browning and Heath-Brown, where they allow the forms to have differing degrees.
- Oscar Marmon: Random Thue and Fermat equations
We consider Thue equations of the form axk+byk = 1, and assuming the truth of the abc-conjecture, we show that almost all locally soluble Thue equations of degree at least three violate the Hasse principle. A similar conclusion holds true for Fermat equations axk+byk+czk = 0 of degree at least six. This is joint work with Rainer Dietmann.